You mixing up 3 things , which should be considered separately:1. Max output current, data usually you can get from IC specification;2. Max Power from power supply what you have;3. Voltage what you need.I say you are on right path on 2-nd, 6 V / 1 A at input could be converted to 12 / 0.5 at the output.(Actually little bit less, as 'd be conversion losses ).

Confused again. So if I had a 12V 0.5A (6W) power source, and I want a converted voltage of 6V. I will have my output at 6V 0.5A? or 6V 1A ?(Power=Voltage x Current = 6W)

The LM317 is a linear regulator. A characteristic of linear regulators is that CURRENT IN is the same as CURRENT OUT. The input and output voltages do not matter. It is how this type of regulator works.

Since current stays contant, the voltage difference from input to output is dropped across the regulator.

Without a proper heatsink, the LM317 with shutdown automatically due to built-in thermal protection at closer to 1A or less not 1.5A. This is one of those "reading the datasheet" things that trips people up... Maximum ratings are not the same as Nominal ratings. You should never PLAN to operate a device at it's Maximum ratings... as those values are there for the designer to know "the point at which you can cause the device to stop working permanently".

Another key point: Physics has some pretty tough laws... and one of them... very oversimplified is: "You never get work done for free." Example: If you start with 6V at 1A power source... and build a circuit that in fact DOES give you 12 Volts out... something has to pay the price... and in this case... it's current availability... you will get 12V Volts... but getting it a the 1 Amp available on the "feed" side is out of the question. You will get reduced current availability in a voltage boost solution.

Something else to keep in mind is that a regulator that is adjustable... is using an external reference that is based on actual regulator output. A fixed regulator has an internal voltage reference that it uses to regulate against.

So, keeping that in mind, if you have a 5V regulator and you feed it 3 volts, the reference circuit never gets to tell the regulator what to regulate against... so regulation of voltage cannot occur.